write the explicit formula for the data.
write a recursive rule for the height of the ball on each successive bounce.
if this ball is dropped from a height of 175 cm, how many times does it bounce before it has a bounce height of less than 8 cm? use the same rebound percentage as in the previous problem.
what is the height of the fourth bounce of this ball if it is dropped from a height of 250 cm? use the same rebound percentage as in the previous problem.

Step-by-step explanation:

This is a geometric sequence so the standard formula for a recursive geometric sequence is

We know the heights and the number of bounces needed to achieve that height, but in order to write the recursive formula we need r.

The value of r is found by dividing each value of a bounce by the one before it.  In other words, bounce 1 divided by the starting height gives a value of r=240/300 so r = .8

Bounce 2 divided by bounce 1: 192/240 = .8

So r = .8

Therefore, the formula is

where

aₙ is the height of the ball after the nth bounce,

a₀ is the starting height of the ball,

.8 is the rebound percentage, and

n-1 is the number of bounces minus 1

The first problem basically asks us to find n when the starting height is 175 and the bounce height is less than 8.  I used 7.  Here is the formula filled in with our info:

and we need to solve for n.  That requires that we take the natural log of both sides.  Here are the steps:

First, divide both sides by 175 to get

Next, take the natural log of both sides:

The power rule of logs says that we can bring the exponent down in front of the log:

Finding the natural logs of those decimals gives us:

Divide both sides by -.223144 to get your n-1 value:

n - 1 = 14.4251067

That means that, since the ball is not bouncing 14.425 times, it bounces 14 times to achieve a height less than 8.  Let's see how much less than 8 by checking our answer.  To do this, we will solve for aₙ when x = 14:

This gives us a height at bounce 14 of 7.697 cm, just under 8!

Now for the next part, we want to use a starting value of 250 and .8 as the rebound height.  We want to find a₄, the height of the 4th bounce.

which simplifies to

Do the math on that to find the height of the 4th bounce from a starting height of 250 cm is 128 cm

First case

   Recursive formula:

   Explicit formula:

Second case: 15 bounces are needed

Third case: 128 cm

Step-by-step explanation:

Let's call h to he height of the ball

From the table, the rate is computed as follows:

r = 240/300 = 192/240 = 153.6/192 =  122.88/153.6 = 98.3/122.88 = 0.8

Which means this is a geometric sequence (all quotients are equal).

Recursive formula:

where n refers to the number of bounces

Explicit formula:

If this ball is dropped from a height of 175 cm, then

If the height must be 8 cm or less:

which means that 15 bounces are needed.

If this ball is dropped from a height of 250 cm, then

For the fourth bounce the height will be:

set each factor equal to 0

4x-3 = 0

2x-1 = 0

solve for x.

4x-3=0

4x=3

x= 3/4

solve for x. 

2x-1= 0

2x= 1

x= 1/2

eventually after a few more steps, your answer should be 1/2 is less than or equal to x and 3/4 is greater than or equal to x